Question

Using ruler and compass only, construct a parallelogram ABCD in which AB = 4 cm, ∠A = 60° and BC = 5 cm.

Answer 1

Given:

AB = 4 cm, BC = 5 cm, ∠A = 60°, construct parallelogram ABCD using ruler and compass only.

Step-wise construction (with reasoning):

1. Draw the base AB of length 4 cm.

Use the ruler to mark segment AB = 4 cm.

2. Construct the ray through A that makes ∠A = 60° with AB.

With centers A and B and radius AB, draw two circles to locate the third vertex E of an equilateral triangle ABE.

Join A to E. The ray AE makes 60° with AB equilateral triangle gives 60° at A.

3. On ray AE mark point D so that AD = 5 cm AD must equal BC.

Open the compass to 5 cm and from A cut off point D on ray AE so that AD = 5 cm.

4. Locate C as the intersection of two circles:

With center B and radius 5 cm to ensure BC = 5, draw a circle.

With center D and radius 4 cm to ensure DC = AB = 4, draw a circle.

Let the appropriate intersection of these two circles be C. Choose the intersection on the same side of AB as D to get the intended shape.

5. Join points: connect C to B and C to D, also join D to A and A to B if not already.

You now have quadrilateral ABCD with AB = 4, BC = 5, AD = 5 and DC = 4.

Reason/Verification:

By construction BC = AD = 5 and AB = DC = 4, so both pairs of opposite sides are equal. A quadrilateral with both pairs of opposite sides equal is a parallelogram; therefore, ABCD is the required parallelogram.